The Born-Oppenheimer approximation

The Born-Oppenheimer approximation

 The electrons are much lighter than the nuclei (m_e/m_H1/1836)  their motion is much faster than the vibrational and rotational motions of the nuclei within the molecule.

 The Schrödinger equation can then be divided into two equations:

1) One describes the motion of the nuclei.

2) The other one describes the motion of the electrons around the nuclei whose positions are fixed.

 The nuclear coordinate R appears as a parameter in the expression of the electronic wave function.

 An electronic wave function ^elect(R,r) and an energy E_elect are associated to each structure of the molecule (set of nuclei coordinates R).

 For each variation of bond length in the molecule (each new R), we can solve the electronic SE and evaluate the energy that the molecule would have in this structure: the molecular potential energy curve is obtained (see Figure).

 The molecule is the most stable (minimum of energy) for one specific position of the nuclei: the equilibrium position R_e.

The electronic Schrödinger equation

D₀

 The zero energy corresponds to the dissociated molecule.

 The depth of the minimum, D_e, gives the bond dissociation energy, D₀, considering the fact that vibrational energy is never zero, but : D₀=D_e-

R

€

12 hω

€

12 hω

Valence-bond theory The hydrogen molecule

A(1) = _H1sA(r1) is the wavefunction of the electron 1 on the 1s orbital of Hydrogen A.

When the atoms are close, it’s not possible to know whether it is electron 1 that is on A or electron 2.

₊= A(1)B(2) + A(2)B(1)

_-= A(1)B(2) - A(2)B(1)

 The system is described by a superposition of the wavefunctions for each possibility: A(1)B(2) and A(2)B(1). Two linear combinations are possible:

₊= A(1)B(2) + A(2)B(1)  between the two nuclei: |₊|²>0  creation of a  bond described by a “bonding molecular orbital”.

_-= A(1)B(2) - A(2)B(1)  between the two nuclei, _- changes sign

 |_-|² =0  creation of a * bond described by a “antibonding molecular orbital”.

Homonuclear diatomic molecules











N 1s

2s

2p

2p

2p

Valence electrons

In N₂: 3 bonds are formed by combining the 3 different 2p orbitals of the 2 nitrogen atoms. This is possible because of the symmetry and the position of those 2p orbitals with respect to each other.

 one  bond and 2  bonds (perpendicular to each other) are formed by spin pairing.

z

y

x

Polyatomic molecules

O 1s

2s

2p

2p

2p

H 1s

H 1s

H₂O: This simple model suggests that the formed angle between the _(O-H) bonds is 90°, as well as their position vs. the paired 2p_x electrons; whereas the actual bond angle is 104.5°.

z

y

x

 C: 1s² 2s² 2p_x¹ 2p_y¹. With the valence bond theory, we expect maximum 2 bonds.

 But, tetravalent carbon atoms are well known: e.g., CH₄. This deficiency of the theory is artificially overcome by allowing for promotion.

Promotion: the excitation of an electron to an orbital of higher energy. This is not what happens physically during bond formation, but it allows to feel the energetics. Indeed, following this artificial excitation, the atom is allowed to create bonds; and consequently, the energy is stabilized… more than the cost of the excitation energy.

 C: 1s² 2s² 2p_x¹ 2p_y¹  1s² 2s¹ 2p_x¹ 2p_y¹ 2p_z¹. CH₄ should be composed of 3 bonds due to the overlap between the 2p of C and the 1s of H, and another  bond coming from the 2s of C and the 1s of H.

 But, it is known that CH₄ has 4 similar bonds. This problem is overcome by realizing that the wavefunction of the promoted atom can be described on different orthonormal basis sets:

1) either the orthogonal hydrogenoic atomic orbitals (AO): 1s, 2s, 2p_x, 2p_y, 2p_z. 2) or equivalently, from another set of orthonormal functions: “the hybrid orbitals”

Promotion

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + −

R r

r - e

= V

B A

1 1 1 4 ₀

2

V πε

H m _e

e

∇ +

= − ^{2 2}

2h

B

A

r_B r_A

e^-

⎟⎠

⎜ ⎞

⎝

⎛

−

− −

− +

S k j R

E e

=

E _{H s}

1 4 ₀

2

1 πε

_-

₊

Energy of the states  and *

H-H⁺: One electron around 2 protons

H  = E 

⎟⎠

⎜ ⎞

⎝

⎛ +

− +

+ +

S k j R

E e

=

E _{H s}

1 4 ₀

2

1 πε

=₊

=_-

Structure of diatomic molecules

Now, we use the molecular orbitals (= ₊ and *= _-) found for the one-electron molecule H-H⁺; in order to describe many-electron diatomic molecules.

The hydrogen and helium molecules

H₂: 2 electrons  ground-state configuration: 1²

He₂: 4 electrons  ground-state configuration: 1² 2*² E

E

E

E

E

< E

stable and does not exist

Increase of electron density

Bond order

Bond order:

b(n-n*)

 The greater the bond order between atoms of a given pair of elements, the shorter is the bond and the greater is the bond strength.

n= number of electrons in the bonding orbital

n*= number of electrons in the antibonding orbital

According to molecular orbital theory,  orbitals are built from all orbitals that have the appropriate symmetry. In homonuclear diatomic molecules of Period 2, that means that two 2s and two 2p_z orbitals should be used. From these four orbitals, four molecular orbitals can be built: 1, 2*, 3, 4*.

Period 2 diatomic molecules

1, 2*, 3, 4*.

With N atomic orbitals  the molecule will have N molecular orbitals, which are combinations of the N atomic orbitals.

dioxygen O

:

The two 2p_x give one _x and one _x* The two 2p_y give one _y and one _y* Bond order = 2

Note: The  orbitals together give rise to an cylindrical distribution of charge. Electrons can circulate around this torus can create magnetic effect detected in NMR

The two last e^- occupy both the

_x* and the _y* in order to decrease their repulsion. The more stable state for 2e^- in different orbitals is a triplet state. O₂ has total spin S=1 (paramagnetic)

Heteronuclear diatomic molecules

 A diatomic molecule with different atoms can lead to polar bond, a covalent bond in which the electron pair is shared unequally by the 2 atoms.

 2 electrons in an molecular orbital composed of one atomic orbital of each atom (A and B).

 = c_A A + c_B B

|c_i|²= proportion of the atomic orbital “i” in the bond Polar bonds

Example: HF

The H1s electron is at higher energy than the F2p orbital.

The bond formation is accompanied with a significant partial negative charge transfer from H to F.

 The situation of covalent polar bonds is between 2 limit cases:

1) The nonpolar bond (e.g.; the homonuclear diatomic molecule): |c_A|²= |c_B|²

2) The ionic bond in A⁺B^- : |c_A|²= 0 and |c_B|²=1

Electronegativity

 A measure of the power of an atom to attract electron to itself when it is part of a compound.

There are different electronegativity definitions, e.g. the Mulliken electronegativity:



=½ (IP + EA)

 IP is the ionization potential = the minimum energy to remove an electron from the ground state of the molecule (Chap 3, p14).

 EA is the electron affinity = energy released when an electron is added to a molecule. EA>0 when the addition of the electron releases energy, i.e. when it stabilizes the molecule.

Electronic transitions Vibrational transitions Rotational

transitions

Selection rules in Raman spectroscopy

Monochromatic incident radiation

E_i= h_i

Scattered radiation

L = lens M = mirror

 Most of the scattered radiation has the same wavelength as the incident beam. This radiation is called the Rayleigh radiation and is the result of elastic scattering.

 About 1 in 10⁷ of the incident photons collide with the molecules, give up some energy, and emerge with a lower energy. These inelastic scattered photons constitute the lower- frequency Stokes radiation from the sample.

 Other incident photons may collect energy from the molecules (if they are already excited), and emerge as higher- frequency anti-Stokes radiation.

General principle

The energy difference between the incident light (E_i) and the Raman scattered light (E_s) is equal to the energy involved in changing the molecule's vibrational state (i.e. getting the molecule to vibrate, E_v). This energy difference is called the Raman shift: E_v = E_i - E_s

Several different Raman shifted signals will often be observed; each being associated with different vibrational (or rotational) motions of molecules in the sample. A plot of Raman intensity vs. Raman shift is a Raman spectrum.

E_v = E_i - E_s R - Rayleigh Scattering

S - Stokes Raman Scattering AS - Anti-Stokes Raman Scattering

0

Virtual level X

Virtual level X

h_inc

h_inc

h_inc

+

-

h_vib

higher-frequency lines: anti-Stokes radiation.

lower-frequency lines: Stokes radiation

 At room temperature, few molecules are in the first excited vibrational levels.

Consequently, the anti-Stokes line have a low intensity.

The vibrational structure: Franck-Condon principle

Classical picture:

Because the nuclei are heavier than electrons, an electronic transition takes place much faster than the nuclei can respond. This is represented by the vertical green arrow in the graph: during the vertical electronic transition, the molecule has the same geometry as before the excitation.

During the transition, the electron density is rapidly built up in new regions of the molecule and removed from others, and the nuclei experience suddenly a new force field, a new potential (upper curve). They respond to this new force by beginning to vibrate.

R_e^* > R_e^gs because an excited state is characterized by an electron in an anti-bonding molecular orbital, which gives rise to an elongation of one or several bonds in the molecule.

E

Separation distance between atoms in the molecule

R

R

*



The fates of electronically excited states

 Nonradiative decay = the excitation energy is transferred into the vibration, rotation, translation of the surrounding molecules via collisions.

 Radiative decay = the excitation energy is discarded as a photon (fluorescence, phosphorescence)

 Dissociation and chemical reaction Molecule A

Molecule B

Collisions

Energy

R

^gs

*_s

v=1v=0 v=2

v=0

R_e^gs R_e^*

Fluorescence

_a _max _b

The excited molecule collides with the surrounding molecules and steps down the ladder of vibrational levels to v=0 of *_s. The surrounding molecules, however, might now be unable to accept the larger energy difference needed to lower the molecule to ^gs. It might therefore survive long enough to undergo spontaneous emission. As a consequence, the transitions in the emission process have lower energy compared to the absorption transition In accord with the Franck-Condom principle, the most probable transition occurs from *_s to the vibrational state of ^gs, for which the molecule has the same inter-atomic separation R_e^*. This vibrational state (v=1 in the Figure) is characterized by a maximum intensity of its vibrational wavefunction at R_e^*. This is the origin of the maximum in the fluorescence or emission spectrum.

Phosphorescence

Conditions:

1) The potential felt by the atoms when the molecule is in its electronic singlet excited state () crosses the potential for the molecule in its triplet excited state ( ). In other words, the structure of the molecule in both states is similar for specific vibrational levels of both states.

2) If there is a mechanism for unpairing two electron spins (and achieving the conversion of  to  ), the molecule may undergo intersystem crossing and becomes in *_T. This is possible if the molecule contains heavy atoms for which spin-orbit coupling is important.

When the molecule reaches the vibrational ground state of *_T, it is trapped!

The solvent cannot absorb the final, large quantum of electronic energy, and the molecule cannot radiate its energy because return to ^gs_S is spin-forbidden….. However, it is not totally spin-forbidden because the spin-orbit coupling mixed the S and T states, such that the transition becomes weakly allowed.

 weak intensity and slow radiative decay (can reach hours!!).

Note: Phosphorescence more efficient for the solid phase

*_S

*_T

^gs_S

Dissociation

A dissociation is characterized by an absorption spectrum composed of two parts:

(i) a vibrational progression (ii) a contiuum absorption

For some molecules, the potential surface of the excited state is strongly shifted to the right compared to the potential of the ground state.

As a consequence, lot of vibrational states of the electronic excited state are accessible (vibrational progression described by the Franck-Condon principle), and the dissociation limit can be reached.

Beyond this dissociation limit, the absorption is continuous because the molecule is broken into two parts. The energy of the photon is used to break a bond and the rest in transformed in the unquantized translational energy of the two parts of the molecule.